Timeline for Lower bound in the singularity of random Bernoulli matrices

6 events

when toggle format	what		by	license	comment
Dec 3 at 21:40	vote	accept	Drew Brady
Dec 3 at 1:24	comment	added	Drew Brady		Thanks. The inclusion-exclusion argument is obvious in retrospect, makes sense!
Dec 3 at 0:09	comment	added	Anthony Quas		The earlier comment gives the elementary argument you're looking for: for any pair, let $E_{i,j}$ be the event that the $i$th and $j$th rows are equal up to signs. Then a lower bound for $\mathbb P(\bigcup_{i<j}E_{i,j})$ is a lower bound for $\mathbb P(\det(A_n)=0)$. You can get a lower bound from the sum of the first two terms of inclusion-exclusion. The contributions to the second term are of the form $E_{i,j}\cap E_{i',j'}$. If $i,i',j,j'$ are distinct, this probability is $2^{-2(n-1)}$. If one of $i',j'$ agrees with one of $i,j$, it's also $2^{-2(n-1)}$. Now sum over $n^4$ many terms.
Dec 2 at 21:05	history	edited	Carlo Beenakker	CC BY-SA 4.0	added 146 characters in body
Dec 2 at 20:07	comment	added	Drew Brady		Yeah, I saw this paper. I guess their interest is in deriving an even sharper asymptotic expansion. However, I was wondering if there are any elementary arguments that lead to the leading order term (at least as a lower bound, obviously we can't expect this for the upper bound, otherwise we'd solve the strong conjecture)
Dec 2 at 19:54	history	answered	Carlo Beenakker	CC BY-SA 4.0